Phil. spacetime // Error Statistics
Eddington on the Deflection of Light
Eddington on the Deflection of Light
Arthur Stanley Eddington was a good writer as well as a good scientist, and his popular book, Space, Time and Gravitation (1920) is still worth reading even today. Since he explains the deflection of starlight (he confirmed this in the1919 British expeditions) so nicely, I cannot refrain from quoting the relevant part (for the following calculation, Schwarzschild coordinates are used, and mass m is measured in units of length. See Schwarzschild Geometry, where M is used instead of m).
The wave-motion in a ray of light can be compared to a succession of long straight waves rolling onward in the sea. If the motion of the waves is slower at one end than the other, the whole wave-front must gradually slew round, and the direction in which it is rolling must change. In the sea this happens when one end of the wave reaches shallow water before the other, because the speed in shallow water is slower. It is well known that this causes waves proceeding diagonally across a bay to slew ound and come in parrallel to the shore; the advanced end is delayed in the shallow water and waits for the other. In the same way when the light waves pass near the sun, the end nearest the sun has the smaller velocity and the wave-front slews round; thus the course of the waves is bent.
Light moves more slowly in a material medium than in vacuum, the velocity being inversely proportional to the refractive index of the medium. The phenomenon of refraction is in fact caused by a slewing of the wave-front in passing into a region of smaller velocity. We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light. To give the velocity 1- 2m/r, the refractive index must be 1/(1 - 2m/r), or, very approximately, 1 + 2m/r. At the surface of the sun, r = 697,000km., m = 1.47 km., hence the necessary refractive index is 1.00000424. At a height above the sun equal to the radius it is 1.00000212.
Any problem on the paths of rays near the sun can now be solved by the methods of geometrical optics applied to the equivalent refracting medium. It is not difficult to show that the total deflection of a ray of light passing at a distance r from the center the sun is (in circular measure)
4m/r,
whereas the deflection of the same ray calculated on the Newtonian theory would be
2m/r.
For a ray grazing the surface of the sun the numerical value of this deflection is
1''.75 (Einstein's theory),
0''.87 (Newtonian theory).
(Eddington 1920, 108-9.)
Recall that Einstein's theory demands spacetime curvature, which means that both space and time are warped. Earlier (in 1911; see Genesis of General Relativity (2)), Einstein considered only the time warps and obtained only a half of the preceding deflection; beginning 1912, he realized that space is also warped, and he envisaged a non-Euclidean geometry for treating gravity. On the other hand, the usual Embedding Diagram shows only the spatial curvature (time is frozen), so that you have to add time warps, in order to obtain the correct curvature. The following figure is an embedding diagram of an equatorial plane of the sun (white part is the interior, and the rest is the exterior space, of the sun).
Try to learn physics or any other disciplines, whenever a related topic appear in the course of reading (that's the best way to do philosophy of science)!
Reference
Eddington, A. S. (1920) Space, Time and Gravitation, Cambridge University Press, 1987.
Last modified Jan. 27, 2003. (c) Soshichi Uchii
